Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.
Then click "Show" button.
You may also see list of all polyomino sets for which data is available here.
Prime rectangles: ≥ 3.
Smallest rectangle (3x9):
Smallest square (9x9):
Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).
Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).
Purple number - prime rectangle (unknown if weakly or strongly prime).
Red number - composite rectangle (which can be divided into two rectangles tileable by this set).
Gray number - it is unknown whether rectangle is prime or composite.
Question mark (?) - solution count is unknown.
Click on underlined numbers to view picture with one solution.
$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)
$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise
$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)
$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$
$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$
$G(N(3); x) = \frac{2x^9}{1 - x^3 - 4x^6 + 2x^9 + 4x^{12}} \tag{1}$
$G(N(6); x) = \frac{4x^9 + 8x^{12}}{1 - 7x^3 + 6x^6 + 32x^9 - 24x^{12} - 48x^{15}} \tag{2}$
$G(N(9); x) = \frac{2x^3 - 10x^6 + 20x^9 + 328x^{12} - 752x^{15} - 1344x^{18} + 4480x^{21} - 6144x^{27}}{1 - 7x^3 - 80x^6 + 360x^9 + 2208x^{12} - 5040x^{15} - 23872x^{18} + 16384x^{21} + 81920x^{24} - 15360x^{27} - 86016x^{30}} \tag{3}$
